Hi there!

It all depends upon your interest as both the institutes are good.

If you are interested in research, metallurgy is a good option. There is good work going on in the field of materials science and engineering abroad. You can diversify to research in materials having applications in electronics, automobiles, power sector or medical sector. You can also work as an engineer in automotive, power sector, steel industries or any metal extraction industry.
If you're not keen on research and you're interested to figure as a plant manager in various industries, chemical engineering is best option in NIT Warangal. There are some common subjects in production engineering and mechanical engineering.
If you are interested to work in government organizations, you have to just check with recent advertisements. Many government organizations have openings for engineering and metallurgical engineering. For those you can't apply with production engineering background. However, You will not face any such problem with private sector.
A few years back, there was no GATE examination for production engineering. The students had to seem in engineering examination. I don't know if this has changed now.
In NIT Trichy all branches are equally good. Lot of companies recruit students. So, if you're looking from that aspect, both branches are fine. There are many co-curricular and additional curricular activities which you'll not only enjoy and learn tons about life generally but also strengthen your resume.